A generalization of the results of Pillai

Yasuhiro Fujita

doi:10.1007/BF00775821

A generalization of the results of Pillai

Yasuhiro Fujita^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-161) showed that the distribution 1-E_α(-x^α), 0<α≤1; 0≤x, where E_α(x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ∞).

Original language	English
Pages (from-to)	361-365
Number of pages	5
Journal	Annals of the Institute of Statistical Mathematics
Volume	45
Issue number	2
DOIs	https://doi.org/10.1007/BF00775821
State	Published - 1993/06

Keywords

Bernstein function
Laplace-Stieltjes transform
Lévy process
geometric infinite divisibility
infinite divisibility

ASJC Scopus subject areas

Statistics and Probability

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10.1007/BF00775821

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title = "A generalization of the results of Pillai",

abstract = "In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-161) showed that the distribution 1-Eα(-xα), 0<α≤1; 0≤x, where Eα(x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ∞).",

keywords = "Bernstein function, Laplace-Stieltjes transform, L{\'e}vy process, geometric infinite divisibility, infinite divisibility",

author = "Yasuhiro Fujita",

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AB - In a recent article Pillai (1990, Ann. Inst. Statist. Math., 42, 157-161) showed that the distribution 1-Eα(-xα), 0<α≤1; 0≤x, where Eα(x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ∞).

KW - Bernstein function

KW - Laplace-Stieltjes transform

KW - Lévy process

KW - geometric infinite divisibility

KW - infinite divisibility

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JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

IS - 2

ER -

A generalization of the results of Pillai

Abstract

Keywords

ASJC Scopus subject areas

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